3.4 Hyperboloidal initial data

Now the obvious problem is to determine hyperboloidal initial data. That such data exist follows already
from Theorem 4 because one can construct hyperboloidal hypersurfaces together with data on them in any
of the radiative solutions whose existence is guaranteed by that theorem. However, one can also construct
such data sets in a similar way to the construction of Cauchy data on an asymptotically Euclidean
hypersurface by solving the constraint equations implied on the Cauchy surface. Let be a hyperboloidal
hypersurface that extends out to in an asymptotically flat vacuum space-time, touching in a
two-surface that is topologically a two-sphere. The assumptions on are equivalent to the fact that
is a smooth Riemannian manifold with boundary, which carries a smooth metric
and a smooth function , obtained by restriction of the unphysical metric and the
conformal factor. The conformal factor is a defining function for the boundary (i.e. it
vanishes only on with non-vanishing gradient), and together with the metric it satisfies

on , where is the metric induced on by the physical metric. Furthermore, let be the
extrinsic curvature of in the physical space-time. Together with it satisfies the vacuum constraint
equations,

where is the Levi–Civita connection of and is its scalar curvature.

Several results of increasing generality have been obtained. We discuss only the simplest case here,
referring to the literature for the more general results. Assume that the extrinsic curvature is a pure trace
term,

The momentum constraint (32) implies that is constant while the hyperboloidal character of
implies that . With these simplifications and a rescaling of with a constant factor, the
Hamiltonian constraint (31) takes the form

A further consequence of the condition (30) is the vanishing of the magnetic part of the Weyl tensor.
For any defining function of the boundary, the conformal factor has the form . Expressing
Equation (33) in terms of the unphysical quantities and yields the single second-order equation

This equation is a special case of the Lichnérowicz equation and is sometimes also referred to as
the Yamabe equation. For a given metric and boundary defining function , it is a
second-order, non-linear equation for the function . Note that the principal part of the equation
degenerates on the boundary. Therefore, on the boundary, the Yamabe equation degenerates to the
relation

Note also that is a solution of Equation (34) which, however, is not useful for our purposes because
it does not give rise to a meaningful conformal factor . Therefore, we require that be
strictly positive and bounded on the boundary. Then the relation above determines the boundary
values of in terms of the function . Taking derivatives of Equation (34), one finds that
also the normal derivative of is fixed on the boundary in terms of the second derivative of
.

A given metric does not fix a unique pair . Therefore, Equation (34) has the property
that, for fixed , rescaling the metric with an arbitrary smooth non-vanishing function on
according to results in a rescaling of the solution of Equation (34) according to
and, hence, a change in the conformal factor .

Now we define the trace-free part of the projection of the trace-free part of
the unphysical Ricci tensor onto , and consider the equations

which follow from the Equations (20) and (16), respectively. Together with the fields , ,
, they provide initial data for all the quantities appearing in the evolution equations under the
given assumptions. As they stand, these expressions are formally singular at the boundary and one needs to
worry about the possibility of a smooth extension of the field to . This question was answered in [5],
where the following theorem was proved:

The functionas well as the tensor fieldsanddetermined on the interiorfromandextend smoothly to all of.

(2)

The conformal Weyl tensor computed from the data vanishes on.

(3)

The conformal class ofis such that the extrinsic curvature ofwith respect to itsembedding inis pure trace.

Condition (3) in Theorem 7 is a weak restriction on the conformal class of the metric on , since
it is only effective on the boundary. It is equivalent to the fact that in the space-time that evolves from the
hyperboloidal data, null-infinity is shear-free. Interestingly, the theorem only requires to be
orientable and does not restrict the topology of any further.

This theorem gives the answer in a highly simplified case because the freedom in the extrinsic curvature
has been suppressed. But there are also several other, less restrictive, treatments in the literature. In [3, 4]
the assumption (30) is dropped, allowing for an extrinsic curvature that is almost general apart from the
fact that the mean curvature is required to be constant. In [101] also this requirement is dropped (but, in
contrast to the other works, there is no discussion of smoothness of the implied conformal initial data), and
in [103] the existence of hyperboloidal initial data is discussed for situations with a non-vanishing
cosmological constant.

The theorem states that one can construct the essential initial data for the evolution once Equation (34)
has been solved. The data are given by expressions that are formally singular at the boundary
because of the division by the conformal factor . This is of no consequences for the analytical
considerations if Condition (3) in Theorem 7 is satisfied. However, even then it is a problem for
the numerical treatments because one has to perform a limit process to get to the values of
the fields on the boundary. This is numerically difficult. Therefore, it would be desirable to
solve the conformal constraints directly. It is clear from Equations (82, 83, 84, 85, 86, 87, 88,
89, 90, 91, 92) that the conformal constraints are regular as well. Some of the equations are
rather simple but the overall dependencies and interrelations among the equations are very
complicated. At the moment there exists no clear analytical method (or even strategy) for solving this
system. An interesting feature appears in connection with Condition (3) in Theorem 7 and
analogous conditions in the more general cases. The necessity of having to impose this condition
seems to indicate that the development of hyperboloidal data is not smooth but in general at
most . If the condition were not imposed then logarithms appear in an expansion of the
solution of the Yamabe equation near the boundary, and it is rather likely that these logarithmic
terms will be carried along with the time evolution, so that the developing null-infinity loses
differentiability. Thus, the conformal boundary is not smooth enough and, consequently, the
Weyl tensor need not vanish on which, in addition, is not necessarily shear-free. The Sachs
peeling property is not completely realized in these situations. One can show [3] that generically
hyperboloidal data fall into the class of “poly-homogeneous” functions, which are (roughly)
characterized by the fact that they allow for asymptotic expansions including logarithmic terms. This
behaviour is in accordance with other work [165] on the smoothness of , in particular with the
Bondi–Sachs type expansions, which were restricted by the condition of analyticity (i.e. no
appearance of logarithmic terms). It is also consistent with the work of Christodoulou and
Klainerman.

Solutions of the hyperboloidal initial value problem provide pieces of space-times that are semi-global in
the sense that their future or past developments are determined depending on whether the hyperboloidal
hypersurface intersects or . However, the domain of dependence of a hyperboloidal initial
surface does not include space-like infinity and one may wonder whether this fact is the reason for the
apparent generic non-smoothness of null-infinity. Is it not conceivable that the possibility of making a
connection between and across to build up a global space-time automatically excludes the
non-smooth data? If we let the hyperboloidal initial surface approach space-like infinity, it might well be
that Condition (3) in Theorem 7 imposes additional conditions on asymptotically flat Cauchy data at
spatial infinity. These conditions would make sure that the development of such Cauchy data is
an asymptotically flat space-time, in particular that it has a smooth conformal extension at
null-infinity.

These questions give some indications about the importance of gaining a detailed understanding of the
structure of gravitational fields near space-like infinity. One of the difficulties in obtaining more information
about the structure at space-like infinity is the lack of examples that are general enough. There exist exact
radiative solutions with boost-rotation symmetry [22]. They possess a part of a smooth null-infinity, which,
however, is incomplete. This is a general problem because the existence of a complete null-infinity with
non-vanishing radiation restricts the possible isometry group of a space-time to be at most one-dimensional
with space-like orbits [14]. Some of the boost-rotation symmetric space-times even have a regular ; thus
they have a vanishing ADM mass. Other examples exist of space-times that are solutions of the
Einstein–Maxwell [39] or Einstein–Yang–Mills [15] equations. They have smooth and complete
null-infinities. However, they were constructed in a way that enforces the field to coincide with the
Schwarzschild or the Reissner–Nordström solutions near . So they are not general enough
to draw any conclusions about the generic behaviour of asymptotically flat space-times near
.

Recently, the still outstanding answer to the question as to whether there exist at all global
asymptotically simple vacuum space-times with smooth null-infinity could be answered in the affirmative.
Corvino [35] has shown that there exist asymptotically flat metrics on with vanishing scalar
curvature that are spherically symmetric outside a compact set. His method of construction
consists of a gluing procedure by which a scalar flat and asymptotically flat metric can be
deformed in an annular region in such a way that it can be glued with a predetermined degree of
smoothness to Schwarzschild data while remaining scalar flat. Very recently a generalisation
of this construction to non-time-symmetric initial data has been presented by Corvino and
Schoen [36].

Such a metric satisfies the constraint equations implied by the vacuum Einstein equations on
a space-like hypersurface of time symmetry. It evolves into a space-time that is identical to
the Schwarzschild solution near space-like infinity. Within this space-time, which exists for
some finite time interval, one can now find hyperboloidal hypersurfaces on which hyperboloidal
initial data are induced. This hypersurface can be chosen close enough to the initial Cauchy
surface so that it intersects the domain of dependence of the asymptotic region of the initial
hypersurface where the data are Schwarzschild. This means that the data implied on the hyperboloidal
hypersurface are also Schwarzschild close to so that the conditions of Theorem 7 are clearly
satisfied.

If one could now apply Friedrich’s stability result (Theorem 6), then the existence of global
asymptotically flat space-times with smooth extension to null-infinity would have been established.
However, one assumption in this theorem is that the data be “sufficiently small”. Therefore, one needs to
check that one can in fact perform Corvino’s construction in the limit of vanishing ADM mass while
maintaining a sufficiently “large” Schwarzschild region around so that one can find enough
hyperboloidal hypersurfaces intersecting the Schwarzschild region in the time development of the data.
Chruściel and Delay [33] (see also the erratum in [32]) have adapted Corvino’s construction to that
situation, and show that it is possible to construct time-symmetric initial data that are Schwarzschild
outside a fixed compact region and with a fixed degree of differentiability in the limit . From
these data one can construct hyperboloidal hypersurfaces of the type required in Theorem 6 on
which hyperboloidal data are induced which satisfy the smallness criterion of that theorem.
Hence, there exists a complete space-time in the future of the initial Cauchy surface that admits
a conformal extension to null-infinity as smooth as one wishes. Since the argument can also
be applied towards the past, one has shown the existence of global space-times with smooth
.